But she doesn't benefit from the Ofsted registration, you do I presume, though government help with child care fees?
If she came to you already registered then you employed her on this basis but if you wanted her to be registered for you own financial gain then she is within her rights to do the first aid ( a condition of the registration) in working hours.
Maybe she feels more comfortable to ask you for time off, than the other family and I can see that you have been very accommodating as an employer, you're feeling taken for granted or like she is having her cake and eating it.
I know for me as a nanny I enjoyed working for two families because I felt in a more powerful position, both families had pros and cons, if you suddenly become uncharacteristically, tough, she may jump ship all together.
I wouldn't have thought that she has more respect for her other employers -she may not have as many working rights, in terms of a contract, with them like she does with you or she may feel uncomfortable to ask them/
I'm sure if you feel she isn't the best fit but you're just putting up with her she has picked up on this and maybe feels less respected by you as a professional.
When do you expect her to do her training= on her days off?
Just giving a nannies POV I hope it's helpful in some way.